# Equation used to obtain the adjustment angle of a pivoting lid support mechanism?

The following image shows a CAD model in its folded and unfolded form. I was able to successfully give it the ability for it to be adjusted between a 25 and 45 degree form, at 5 degree increments; but would still like to know a mathematical equation, for future projects.
The following images displays the arms at various angles: This is the pivot bracket that moves to hold the arm at each adjusted angle, it is rotated at the center of the small sky-blue circle; please note the green circle has a triangle within it as well: One end of the pivot bracket slides along 5mm deep notch/track/mortice, that is sloped at 2.5 degree angle: This the approximate distance of each holes: I’ve gone ahead and measured each internal angles and the distance between the arm & pivot bracket at each adjusted angle; they each add up to 180 degrees: I’m interested in the mathematical formula that would be used to obtain the values: 29.1, 17.0, 15.0, 14.1

• How long it took you to draw these. Looks very impressive. Jan 16 '19 at 18:06

This is an impressive amount of work but unfortunately the CAD drawings don't seem to match the assumption in drawing the triangles. From the side views, The locking wheel moves along a line that does not pass through the pivot point.

But ignoring that, and taking the 45 degree case as "exact" with the support vertical, Pythagoras' theorem says that the length of the sloping arm is $$\sqrt{2}$$ times the length of the support.

For the other angles, you can use the sine rule to find the other angle at the base of the triangle. If the right-hand angle is $$x$$, the left hand one is $$\sin^{-1}(\sqrt 2 \sin x)$$.

That gives the angles as

25 36.7
30 45 (exactly)
35 54.2
40 65.3
45 90 (exactly)


These aren't quote the same as yours, probably for the reason I gave at the start of the answer.

You can then calculate the third angle by subtracting the others from 180 degrees:

25 36.7 118.3
30 45   105
35 54.2  90.8
40 65.3  74.7
45 90    45


I don't know what your angles of 29.1 etc are supposed to be.

You can calculate the base of the triangles using the sine rule again. The calculated value for the 25 degree case would be $$54.45\, \sin(118.3)\, / \sin(25) = 111.35$$.

angle base-length
45     53.45
40     80.09
35     93.18
30    103.26
25    111.35


I don't know what your lengths are measuring. They are obviously not the base of the triangles, because they don't consistently get longer as the angle reduces from 45 to 25.